Formulations of General Relativity
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Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
Stanisław Zaremba
Danuta Ciesielska*, Krzysztof Ciesielski** Instytut Historii Nauki im. L. i A. Birkenmajerów, PAN Warszawa Instytut Matematyki, Wydział Matematyki i Informatyki, UJ Kraków SSW B (1–192) O DLO K KRTKIE PRZEDSTAWIENIE POLSKIEJ I KRAKOWSKIEJ MATEMATYKI DO POCZTKW XX WIEKU Znaczący rozwój matematyki na świecie datuje się na drugą połowę drugiego tysiąclecia n.e.; przez poprzedzające go półtora tysiąclecia uzyskiwane wyniki były skromne w porównaniu z tym, co uzyskano później. Jednakże do początku XX wieku Polska była z dala od europejskiej, a tym bardziej światowej czołówki. Potęgami ma- tematycznymi były Francja i Niemcy. Ważne rezultaty osiągano w Wielkiej Brytanii i we Włoszech. Sporadycznie pojawiali się słynni matematycy także w innych krajach, jednak w podręcznikach historii matematyki trudno znaleźć wśród nich Polaków. Od XIV wieku Polska miała się czym poszczycić naukowo. Akademia Krakow- ska była drugim uniwersytetem powstałym w środkowej Europie, od początku wy- kładano tu przedmioty kojarzone z matematyką. Na początku XV wieku krakowski mieszczanin Jan Stobner ufundował specjalną katedrę matematyki i astronomii; druga katedra związana z matematyką została ufundowana przez Marcina Króla z Żurawi- cy (ok.1422–ok.1460), pół wieku później. Był to jednak okres istotnie poprzedzający czasy większych osiągnięć matematycznych. Pewne osiągnięcia matematyczne miał ponad wiek później Mikołaj Kopernik (1473–1543), jego nazwisko kojarzone jest jed- nak (oczywiście słusznie) głównie z astronomią. Dopiero w XVII wieku pojawił się w Krakowie matematyk europejskiego formatu – Joannes Broscius (1585–1652; Jan Brożek, znany też jako Brzozek). Był nie tylko mate- matykiem, ale też flozofem, astronomem, teologiem, lekarzem i historykiem nauki. Ma on na swoim koncie znaczące osiągnięcia, głównie związane są z teorią liczb. -
Variational Problem and Bigravity Nature of Modified Teleparallel Theories
Variational Problem and Bigravity Nature of Modified Teleparallel Theories Martin Krˇsˇs´ak∗ Institute of Physics, University of Tartu, W. Ostwaldi 1, Tartu 50411, Estonia August 15, 2017 Abstract We consider the variational principle in the covariant formulation of modified telepar- allel theories with second order field equations. We vary the action with respect to the spin connection and obtain a consistency condition relating the spin connection with the tetrad. We argue that since the spin connection can be calculated using an additional reference tetrad, modified teleparallel theories can be interpreted as effectively bigravity theories. We conclude with discussion about the relation of our results and those obtained in the usual, non-covariant, formulation of teleparallel theories and present the solution to the problem of choosing the tetrad in f(T ) gravity theories. 1 Introduction Teleparallel gravity is an alternative formulation of general relativity that can be traced back to Einstein’s attempt to formulate the unified field theory [1–9]. Over the last decade, various modifications of teleparallel gravity became a popular tool to address the problem of the accelerated expansion of the Universe without invoking the dark sector [10–38]. The attractiveness of modifying teleparallel gravity–rather than the usual general relativity–lies in the fact that we obtain an entirely new class of modified gravity theories, which have second arXiv:1705.01072v3 [gr-qc] 13 Aug 2017 order field equations. See [39] for the extensive review. A well-known shortcoming of the original formulation of teleparallel theories is the prob- lem of local Lorentz symmetry violation [40, 41]. -
Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md
International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds. -
Advanced Lectures on General Relativity
Lecture notes prepared for the Solvay Doctoral School on Quantum Field Theory, Strings and Gravity. Lectures given in Brussels, October 2017. Advanced Lectures on General Relativity Lecturing & Proofreading: Geoffrey Compère Typesetting, layout & figures: Adrien Fiorucci Fonds National de la Recherche Scientifique (Belgium) Physique Théorique et Mathématique Université Libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium Please email any question or correction to: [email protected] Abstract — These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as con- served quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions. Table of contents 1 Surface charges in Gravitation ................................... 7 1.1 Introduction : general covariance and conserved stress tensor..............7 1.2 Generalized Noether theorem................................. 10 1.2.1 Gauge transformations and trivial currents..................... 10 1.2.2 Lower degree conservation laws........................... 11 1.2.3 Surface charges in generally covariant theories................... 13 1.3 Covariant phase space formalism............................... 14 1.3.1 Field fibration and symplectic structure....................... 14 1.3.2 Noether’s second theorem : an important lemma................. 17 Einstein’s gravity.................................... 18 Einstein-Maxwell electrodynamics.......................... 18 1.3.3 Fundamental theorem of the covariant phase space formalism.......... 20 Cartan’s magic formula............................... -
FROM DIFFERENTIATION in AFFINE SPACES to CONNECTIONS Jovana -Duretic 1. Introduction Definition 1. We Say That a Real Valued
THE TEACHING OF MATHEMATICS 2015, Vol. XVIII, 2, pp. 61–80 FROM DIFFERENTIATION IN AFFINE SPACES TO CONNECTIONS Jovana Dureti´c- Abstract. Connections and covariant derivatives are usually taught as a basic concept of differential geometry, or more precisely, of differential calculus on smooth manifolds. In this article we show that the need for covariant derivatives may arise, or at lest be motivated, even in a linear situation. We show how a generalization of the notion of a derivative of a function to a derivative of a map between affine spaces naturally leads to the notion of a connection. Covariant derivative is defined in the framework of vector bundles and connections in a way which preserves standard properties of derivatives. A special attention is paid on the role played by zero–sets of a first derivative in several contexts. MathEduc Subject Classification: I 95, G 95 MSC Subject Classification: 97 I 99, 97 G 99, 53–01 Key words and phrases: Affine space; second derivative; connection; vector bun- dle. 1. Introduction Definition 1. We say that a real valued function f :(a; b) ! R is differen- tiable at a point x0 2 (a; b) ½ R if a limit f(x) ¡ f(x ) lim 0 x!x0 x ¡ x0 0 exists. We denote this limit by f (x0) and call it a derivative of a function f at a point x0. We can write this limit in a different form, as 0 f(x0 + h) ¡ f(x0) (1) f (x0) = lim : h!0 h This expression makes sense if the codomain of a function is Rn, or more general, if the codomain is a normed vector space. -
General Relativity
GENERALRELATIVITY t h i m o p r e i s1 17th April 2020 1 [email protected] CONTENTS 1 differential geomtry 3 1.1 Differentiable manifolds 3 1.2 The tangent space 4 1.2.1 Tangent vectors 6 1.3 Dual vectors and tensor 6 1.3.1 Tensor densities 8 1.4 Metric 11 1.5 Connections and covariant derivatives 13 1.5.1 Note on exponential map/Riemannian normal coordinates - TO DO 18 1.6 Geodesics 20 1.6.1 Equivalent deriavtion of the Geodesic Equation - Weinberg 22 1.6.2 Character of geodesic motion is sustained and proper time is extremal on geodesics 24 1.6.3 Another remark on geodesic equation using the principle of general covariance 26 1.6.4 On the parametrization of the path 27 1.7 An equivalent consideration of parallel transport, geodesics 29 1.7.1 Formal solution to the parallel transport equa- tion 31 1.8 Curvature 33 1.8.1 Torsion and metric connection 34 1.8.2 How to get from the connection coefficients to the connection-the metric connection 34 1.8.3 Conceptional flow of how to add structure on our mathematical constructs 36 1.8.4 The curvature 37 1.8.5 Independent components of the Riemann tensor and intuition for curvature 39 1.8.6 The Ricci tensor 41 1.8.7 The Einstein tensor 43 1.9 The Lie derivative 43 1.9.1 Pull-back and Push-forward 43 1.9.2 Connection between coordinate transformations and diffeomorphism 47 1.9.3 The Lie derivative 48 1.10 Symmetric Spaces 50 1.10.1 Killing vectors 50 1.10.2 Maximally Symmetric Spaces and their Unique- ness 54 iii iv co n t e n t s 1.10.3 Maximally symmetric spaces and their construc- tion -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
The Category of Affine Connection Control Systems 3
Proceedings of the 39th IEEE Conference on Decision and Control pages 5119{5124, December 2000 doi: 10.1109/CDC.2001.914762 2000 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE The category of affine connection control systems∗ Andrew D. Lewisy 2000/02/28 Last updated: 2003/07/23 Abstract The category of affine connection control systems is one whose objects are control systems whose drift vector field is the geodesic spray of an affine connection, and whose control vector fields are vertical lifts to the tangent bundle of vector fields on configu- ration space. We investigate morphisms (feedback transformations) in this category. Keywords. control of mechanical systems, affine connections AMS Subject Classifications (2020). 53B05, 70Q05, 93B29 1. Introduction It is apparent that the study of what we will in this paper call “affine connection control systems" has a significant r^oleto play in the field of mechanical control systems. In a series of papers, [e.g., Bullo, Leonard, and Lewis 2000, Lewis 1998, Lewis 1999, Lewis and Murray 1997a, Lewis and Murray 1997b], the author and various coauthors have shown how the affine connection framework is useful in looking at mechanical systems whose Lagrangian is the kinetic energy with respect to a Riemannian metric, possibly in the presence of constraints linear in velocity [e.g., Lewis 1997, Lewis 2000]. -
Weyl's Spin Connection
THE SPIN CONNECTION IN WEYL SPACE c William O. Straub, PhD Pasadena, California “The use of general connections means asking for trouble.” —Abraham Pais In addition to his seminal 1929 exposition on quantum mechanical gauge invariance1, Hermann Weyl demonstrated how the concept of a spinor (essentially a flat-space two-component quantity with non-tensor- like transformation properties) could be carried over to the curved space of general relativity. Prior to Weyl’s paper, spinors were recognized primarily as mathematical objects that transformed in the space of SU (2), but in 1928 Dirac showed that spinors were fundamental to the quantum mechanical description of spin—1/2 particles (electrons). However, the spacetime stage that Dirac’s spinors operated in was still Lorentzian. Because spinors are neither scalars nor vectors, at that time it was unclear how spinors behaved in curved spaces. Weyl’s paper provided a means for this description using tetrads (vierbeins) as the necessary link between Lorentzian space and curved Riemannian space. Weyl’selucidation of spinor behavior in curved space and his development of the so-called spin connection a ab ! band the associated spin vector ! = !ab was noteworthy, but his primary purpose was to demonstrate the profound connection between quantum mechanical gauge invariance and the electromagnetic field. Weyl’s 1929 paper served to complete his earlier (1918) theory2 in which Weyl attempted to derive electrodynamics from the geometrical structure of a generalized Riemannian manifold via a scale-invariant transformation of the metric tensor. This attempt failed, but the manifold he discovered (known as Weyl space), is still a subject of interest in theoretical physics. -
GEOMETRIC INTERPRETATIONS of CURVATURE Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Tran
GEOMETRIC INTERPRETATIONS OF CURVATURE ZHENGQU WAN Abstract. This is an expository paper on geometric meaning of various kinds of curvature on a Riemann manifold. Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Transport 3 4. Geodesics and the Exponential Map 4 5. Riemannian Curvature Tensor 5 6. Taylor Expansion of the Metric in Normal Coordinates and the Geometric Interpretation of Ricci and Scalar Curvature 9 Acknowledgments 13 References 13 1. Notation and Summation Conventions We assume knowledge of the basic theory of smooth manifolds, vector fields and tensors. We will assume all manifolds are smooth, i.e. C1, second countable and Hausdorff. All functions, curves and vector fields will also be smooth unless otherwise stated. Einstein summation convention will be adopted in this paper. In some cases, the index types on either side of an equation will not match and @ so a summation will be needed. The tangent vector field @xi induced by local i coordinates (x ) will be denoted as @i. 2. Affine Connections Riemann curvature is a measure of the noncommutativity of parallel transporta- tion of tangent vectors. To define parallel transport, we need the notion of affine connections. Definition 2.1. Let M be an n-dimensional manifold. An affine connection, or connection, is a map r : X(M) × X(M) ! X(M), where X(M) denotes the space of smooth vector fields, such that for vector fields V1;V2; V; W1;W2 2 X(M) and function f : M! R, (1) r(fV1 + V2;W ) = fr(V1;W ) + r(V2;W ), (2) r(V; aW1 + W2) = ar(V; W1) + r(V; W2), for all a 2 R. -
The Affine Connection Structure of the Charged Symplectic 2-Form(1991)
On the Affine Connection Structure of the Charged Symplectic 2-Form† L. K. Norris‡ ABSTRACT It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion– free and the anti-symmetric part of the R4∗ translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravi- tational and electromagnetic fields can therefore be reformulated as a P (4) = O(1, 3)⊗R4∗ geometric theory with phase space the affine cotangent bundle AT ∗M of spacetime. The source-free Maxwell equations are reformulated as a pair of geometrical conditions on the R4∗ curvature that are exactly analogous to the source-free Einstein equations. †International Journal of Theoretical Physics, 30, pp. 1127-1150 (1991) ‡Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205 1 1. Introduction The problem of geometrizing the relativistic classical mechanics of charged test par- ticles in curved spacetime is closely related to the larger problem of finding a geometrical unification of the gravitational and electromagnetic fields. In a geometrically unified theory one would expect the equations of motion of classical charged test particles to be funda- mental to the geometry in a way analogous to the way uncharged test particle trajectories are geometrized as linear geodesics in general relativity. Since a satisfactory unified theory should contain the known observational laws of mechanics in some appropriate limit, one can gain insight into the larger unification problem by analyzing the geometrical founda- tions of classical mechanics.